Contextualization

Hello, students! Our journey now takes us to the universe of Algebra, more specifically, we will delve into the study of quadratic functions and more specifically, into their graph. For many students, the quadratic function is one of the most significant first experiences with analytic geometry, which is the area of mathematics that works on integrating Algebra and Geometry.

The quadratic function is represented by the formula f(x) = ax^2 + bx + c and has graphs called parabolas. The position and shape of these parabolas depend on the coefficients a, b, and c in the equation. The graph will allow us to extract various information about the function, such as the vertex of the parabola, the roots, the concavity, and, naturally, the relationship between x and y.

Importance and Applications in the Real World

The quadratic function is more than just a simple algebra topic. It has a wide range of applications in different areas, such as Civil Engineering, Economics, Chemistry, Physics, and many others.

In engineering, for example, it can be used in the structural calculation of bridges and buildings. In economics, it assists in cost analysis and profit optimization. In physics, it is essential to understand the trajectory of objects thrown in the air (parabolic motion). And in chemistry, it is used in the study of chemical reactions.

In this project, our focus will be on understanding the quadratic function and how to create its graph. Through this activity, you will be able to see the function in a more practical way and also apply it to real-world problems.

Practical Activity

Activity Title: Designing Quadratic Functions Graphs

Project Objective:

This project aims to deepen the knowledge acquired about second-degree polynomial functions, specifically about the graphical representation of these functions. The project seeks to develop not only the students' mathematical skills but also their competencies in teamwork, time management, solving complex problems, and creative thinking.

Detailed Project Description:

In a practical and engaging way, the activity proposes the creation of a visual catalog that contemplates the various possible configurations of quadratic function graphs. Each group of students should elaborate a 'Parabolas Atlas', composed of graphs of different quadratic functions, highlighting their main characteristics and associating each function with a practical everyday problem.

The functions should vary in terms of roots (real and complex, repeated and distinct), concavity (upward and downward), and displacements (on the x or y-axis). In addition, each group should create a practical problem for each function presented, demonstrating in a creative way how quadratic functions appear in everyday situations.

Required Materials:

1. Computer with internet access and a graph plotting software, such as GeoGebra, Desmos, or even Google Sheets.
2. Notebook or paper for notes and calculations.
3. Time and willingness for group discussions, research, and study.

Activity Step-by-Step:

Step 1: Group Formation - Students should divide into groups of 3 to 5 members.

Step 2: Research and Planning - Each group should research examples of everyday situations that can be modeled by quadratic functions and plan their Parabolas Atlas. Each selected function must be associated with a practical example.

Step 3: Drawing the Graphs - Using a graph plotting software, the groups must generate each of the planned graphs, noting their main characteristics.

Step 4: Elaboration of the Parabolas Atlas - In a text document (Word, Google Docs, etc), the groups must elaborate their Atlas, containing: - The graph of each function, - A description of its characteristics (roots, concavity, displacements), - The associated practical situation, - The mathematical development of the function.

Step 5: Report Writing - Each group must write a report containing:

1. Introduction: Contextualization of the quadratic function and its importance, as well as the project's objectives.
2. Development: Explanation of the theory of quadratic functions, detailing the activities carried out, presentation and discussion of the obtained graphs.
3. Conclusion: Recap of the main points, presentation of the learnings obtained, and conclusions about the project.
4. Bibliography: References of the materials consulted for the project.

Step 6: Project Presentation - Each group will present their Parabolas Atlas to the class, discussing the learning journey they went through, demonstrating the graphs, and discussing the practical situations.

The project duration is one month, with an individual dedication of approximately five to ten hours.

The project deliverables will be the Parabolas Atlas and the associated report. To complete the project, it is essential to delve into the concepts of the quadratic function and creatively explore its real-world applications. Additionally, we highlight the value of teamwork and dialogue, which are skills valued not only in Mathematics but in all areas of life.